On a warm afternoon in Liberty City, three women stood behind a table offering a smattering of fresh produce: 10 tomatoes, five cucumbers, three peppers, five mangoes, one cabbage and a large box of green beans.

They represented a nonprofit, Urban GreenWorks, dedicated to bringing fresh produce to poor neighborhoods with the idea that better food choices will help reverse the wave of obesity sweeping America, particularly devastating in low-income areas.

Though their table was set up on the shaded sidewalk on busy Northwest 62nd Street, not a single customer approached the stand over a 30-minute period.

Meanwhile, two blocks away at the Monar Market, Ronnie Othman was ringing up the cash register about every 30 seconds: Fritos and a Coke, a packaged cookie and Seven Up, Doritos and a Coke. The convenience store offered some apples and bananas in a basket, and a lone head of lettuce in a refrigerated case. “But they don’t sell all the time,” Othman said.

Studies by the Centers for Disease Control show that all of America is getting fatter -- rich and poor, all ethnic groups, all education levels. Still, the poor are more likely to be obese than the affluent. First Lady Michelle Obama and many nutritional activists believe that one reason the poor tend to be fat is that they often live in “food deserts,” places without large supermarkets offering fresh fruits and vegetables available in more affluent neighborhoods.

That theory is questioned by many researchers. “The idea of food deserts has nothing to do with obesity,” says Roland Sturm, a Rand Institute researcher who co-authored a recent paper on a study of 13,000 children that found no correlation between kids’ obesity and access to fresh produce. A much better predictor, he says: parents’ weight.

“You’re just not going to change behavior by offering more vegetables,” said Sturm in a telephone interview. “You can see people who shop at Whole Foods are thinner, but that doesn’t mean that if you had a Whole Foods on every corner, everybody would be thinner.”

Sturm’s work is backed up by a 2011 study in the Archives of Internal Medicine, which followed 5,100 people over 15 years and found that greater supermarket availability had no correlation with how frequently people ate fruits and vegetables.

Still, major efforts nationally and locally are focused on making healthy foods available to the poor. The Obama administration in 2009 targeted $370 million in stimulus funds for a program called Communities Putting Prevention to Work, with much of the money intended to fight obesity. About $14.7 million of the money went to Miami-Dade County to reduce obesity through initiatives that included getting more nutritional food into public schools and creating bike paths to promote exercise.

The Miami-Dade Health Department received $2.7 million to fund 14 initiatives, including hiring a dozen employees, sponsoring wellness seminars and funding a 16-page report, released in March, called “Expanding Supermarket Access in Areas of Need,” with a map showing large sections of northwestern Miami-Dade — particularly parts of Liberty City, Opa-locka and Hialeah — as “areas of greatest need.”

Health Department Administrator Lillian Rivera wrote in the report that “research shows that better access to affordable, nutritious food is associated with healthier eating habits.’’ She noted that “250,000 Miami-Dade residents (10 percent) live in low-income areas that have poor supermarket access and higher than average death rates from diet-related causes.”

Rivera said recently that, despite questions about the food desert concept, she still believes “providing access to healthier options will help to decrease the obesity epidemic, but it must be part of a multifaceted approach,” including healthier options in schools, childcare centers and worksites.

Miami-Dade’s report is the first step toward bringing fresh produce to poor areas, says Karen Weller of the Health Department. She says her group is working with small grocers to find ways to encourage them to offer more vegetables, building on a program in Philadelphia that has been praised for its extensive work with small stores, sometimes providing subsidized refrigeration units to preserve the vegetables or setting up “healthy snack shelves” near the cash registers.

The Health Department also has invested $280,000 to subsidize seven farmers’ markets to serve poor neighborhoods. Some were seasonal and have already closed for the summer. Others seem to have less than reliable hours. Although a website that Weller said showed the locations and times of the markets, at least one, in Brownsville, wasn’t open when the website said it would be.

James Jiler, executive director of Urban GreenWorks, which runs the Liberty City market and has not received stimulus funding, called the Health Department farmers’ market subsidies “a huge boondoggle,” because the grant money hadn’t been distributed until December, so the markets didn’t open till January. Then the funding stopped in March when stimulus grants ended.

Rivera, the Health Department administrator, says the funds “were spent wisely. They were spent strategically, and they were spent on evidence-based practice.”

Karen Landry, executive director of War on Poverty-Florida, a nonprofit based in Jacksonville , which received $35,000 for an Opa-locka farmers’ market, says even though the market was only open for five months, it served as a spark. “The community is being engaged in developing enduring solutions. … We were very satisfied with the support that we received.” Landry said they plan to reopen the market in September.

One market that has continued is in downtown Miami, in the courtyard of the Stephen P. Clark Government Center, which is on the edge of Overtown. Run by Earth Learning, the market in early May had a light but steady flow of customers who came for cucumbers, eggplant, custard apples, arugula and wheat grass, a delicacy that goes for $12 a pound.

Most of the customers were middle-class government employees or those visiting the area. Lydia Mackie, who was operating the stand, said few poor people from Overtown came by. She estimated that she had “two to six” food stamp customers that she called “regulars” who shop there biweekly.

Jiler, head of Urban GreenWorks, acknowledges that the poor don’t automatically flock to farmer’s markets. “It is slow. We are learning. You have to change eating habits. … We have a whole generation that is brought up on fast food.”

But Urban GreenWorks sometimes has problems of its own. On an April afternoon when the nonprofit said it was supposed to be offering a market at old Opa-locka city hall, there was no sign of it. The director of projects and programming, Roger Horne, later said the food hadn’t been delivered.

A half-block away from the city hall is Diaz Supermarket, a large store that occupies a former Pantry Pride location. It offered beans, cucumbers, peppers, carrots, tomatoes and oranges — at prices just slightly higher than Winn-Dixie and Publix.

Owner John Diaz said he didn’t understand why anyone would consider the area a food desert that needed a farmer’s market. “It’s like going to a Home Depot parking lot and selling plywood.”

That holds true, too, for Liberty City. Thirteen blocks from the lonely tables of GreenWorks on Northwest 62nd Street is a huge, recently remodeled Winn-Dixie with a large produce department.

“We like to be a pleasant surprise,” said store manager James Morris, who says the store also contributes to a local food bank.

While managers say that Winn-Dixie is profitable, grassroots efforts in Miami’s poor neighborhoods remain iffy, especially when they rely on government support. An Overtown vegetable garden on Northwest Third Avenue flourished for three years — until it lost $100,000 a year in support from the city of Miami.

Organizer Marvin Dunn says it was “a forlorn hope” that Overtown residents could maintain the garden as a unsubsidized source of food, but he hoped to get new funding for an agribusiness venture offering jobs to local residents and selling produce to large purchasers.

On the national level, the Obama administration remains strongly committed to bringing fresh produce to poor neighborhoods. Although Congress last year rejected an administration request for $400 million to help supermarkets set up operations in lower-income areas, White House spokeswoman Joanna Rosholm says the administration “continues to work to ensure that all families have access to healthy and affordable food.”

Last year, the Treasury Department provided $25 million in grants to fund healthy food initiatives and a tax credit program was expected to generate over $400 million worth of activities to improve food access, Rosholm said.

The administration has also funded an interactive map through the U.S. Department of Agriculture that shows census tracts that qualify as food deserts. In South Florida, large areas of Opa-locka and Hialeah fit that category, as well as parts of Biscayne Park, Miramar, Pembroke Pines and a section near Northeast 79th Street and Biscayne Boulevard.

Several local nonprofits are also involved in the obesity battle, led by the Blue Cross and Blue Shield of Florida Foundation, which has put almost $900,000 into anti-obesity campaigns in Hialeah and Opa-locka since 2008, including parent cooking classes, working to change a bus route to put it closer to a supermarket and helping build a recreation center in Opa-locka — “a safe place to help them get healthy,” says foundation vice president Susan Towler.

Sturm, the Rand obesity expert, calls the efforts in poor neighborhoods “well intended” but “really misguided.” He believes the attention should be paid to the larger issue, such as super-sized fast foods, to deal with the fact that all of America, rich and poor, is getting fatter.